Perimeter of Compound Shapes –

Perimeter of compound shapes is a frequently tested GCSE Maths topic at both Foundation and Higher tiers. A compound shape is made by combining two or more simple shapes such as rectangles, triangles, semicircles, or quarter circles. The perimeter is the total distance around the outside of the shape. This guide shows you how to find missing lengths, identify every outer edge, and handle curved sections accurately.

What Is the Perimeter of a Compound Shape?

The perimeter is the total length of all the outer edges of a shape. For compound shapes you must:

1. Identify which edges are on the outside (some internal edges are not part of the perimeter).
2. Work out any missing lengths using the given dimensions.
3. Add all outer edges together, including any curved sections.

Key Formulas

When a semicircle or quarter circle is attached to a straight shape, the diameter or radius edge usually replaces a straight edge — so it is not counted separately in the perimeter.

Step-by-Step Method

1. Label all given dimensions on the diagram.
2. Find missing lengths — in L-shapes and T-shapes, subtract known lengths from totals to find the gaps.
3. Identify every outer edge — trace your finger around the outside of the shape, listing each edge length.
4. Calculate curved edges — use πr for semicircles or πr/2 for quarter circles.
5. Add all outer edges together to get the total perimeter.

Worked Example 1 — Foundation Level

Question: An L-shaped room has outer dimensions 10 m by 8 m. A 4 m by 3 m rectangle is cut from the top-right corner. Find the perimeter.

Working:

Step 1 — The outer edges along the bottom = 10 m. The left side = 8 m.

Step 2 — The top edge goes 10 − 4 = 6 m across, then drops 3 m, then goes 4 m across to the right, then up 8 − 3 = 5 m... but wait — trace the outside carefully.

Step 3 — Going clockwise from bottom-left: bottom = 10 m, right side = 8 − 3 = 5 m, step inward = 4 m, step down = 3 m, top = 10 − 4 = 6 m, left side = 8 m.

Step 4 — Perimeter = 10 + 5 + 4 + 3 + 6 + 8 = 36 m.

Answer: The perimeter is 36 m.

Worked Example 2 — Higher Level

Question: A shape consists of a rectangle 12 cm by 6 cm with a semicircle attached to one of the shorter ends. Find the perimeter. Give your answer to 1 decimal place.

Working:

Step 1 — The semicircle is attached to a 6 cm edge, so its diameter = 6 cm and radius = 3 cm.

Step 2 — The straight edges forming the perimeter: two long sides of 12 cm each, plus one short side of 6 cm. The other short side is replaced by the semicircle arc.

Step 3 — Semicircle arc = πr = π × 3 = 3π cm.

Step 4 — Perimeter = 12 + 6 + 12 + 3π = 30 + 9.4248 = 39.4 cm (1 d.p.).

Answer: The perimeter is 39.4 cm.

Worked Example 3 — Exam Style

Question: A running track consists of a rectangle 100 m by 40 m with a semicircle on each short end. Find the total perimeter of the track to 1 decimal place.

Working:

Step 1 — Each semicircle has diameter 40 m, so radius = 20 m.

Step 2 — Two semicircle arcs together form a complete circle: circumference = 2πr = 2 × π × 20 = 40π m.

Step 3 — The two long straight sides contribute 100 + 100 = 200 m. The short sides are replaced by the semicircles.

Step 4 — Total perimeter = 200 + 40π = 200 + 125.664 = 325.7 m (1 d.p.).

Answer: The perimeter is 325.7 m.

Common Mistakes

• Including internal edges. When two shapes are joined, the shared edge is inside the compound shape and must not be counted in the perimeter.
• Forgetting the straight diameter edge of a semicircle. If a semicircle sits on top of a shape and its diameter is exposed (not shared), you must add the diameter as well as the arc.
• Using diameter instead of radius in arc formulas. Semicircle arc = πr, not πd. Always halve the diameter first.

Exam Tips

• Trace around the outside of the shape with your pen to make sure you count every edge exactly once.
• For L-shapes and T-shapes, the perimeter is often equal to the perimeter of the enclosing rectangle — check this shortcut.
• Always state the formula you are using for curved sections to earn method marks.
• Give your answer in the units specified and round only at the final step.

Practice Questions

Q1 (Foundation): A T-shape has a horizontal bar 14 cm by 3 cm on top of a vertical bar 4 cm by 8 cm (centred). Find the perimeter.

Q2 (Foundation): A rectangle 8 cm by 5 cm has a quarter circle of radius 5 cm cut from one corner. Find the perimeter to 1 d.p.

Q3 (Higher): A shape is made of a rectangle 20 cm by 10 cm with semicircles on both short ends. Find the exact perimeter.

Practise compound shape perimeter questions with instant feedback free on GCSEMathsAI.

Related Topics

• Perimeter — perimeter of simple shapes.
• Area of 2D Shapes — areas of compound shapes.
• Arc Length and Sector Area — arc lengths for curved sections.

Summary

Perimeter of compound shapes requires you to trace the entire outside boundary, find missing lengths by subtraction, and use circle formulas for any curved sections. The most common errors are including internal edges and using the wrong formula for semicircular arcs. Always label dimensions, trace the outline systematically, and verify your answer by checking that every outer edge has been counted exactly once.